The lamp and the floor cannot both be recovered

A shadow can be un-cast ran a point lamp’s projection backwards: given the shadow and the lamp, the caster’s own outline comes back exactly, because a point light’s projection is a plane projectivity and every plane projectivity has an inverse. The floor is a choice of coordinates then asked what happens once the receiver is not flat, and found that a homology fitted to a curved floor returns a fitted “curvature” with no referent at all whenever the true departure from flat is something other than curvature. Both essays assumed the lamp’s position was already known. This essay removes that assumption and asks the question those two were quietly standing on: given only the photograph — the shadow and the card that cast it — are the lamp and the floor separately recoverable at all?

They are not, and the reason is sharper than “the arithmetic is hard.” There exists a continuous, one-parameter family of lamp-and-floor pairs, all different from each other, every one of which casts the identical shadow of the identical card — not approximately, but to a fraction of a thousandth of a pixel — because the map from one member of the family to any other is a homology centred on the camera’s own eye, and a homology centred on the eye moves points without moving anything the eye sees. A single photograph cannot tell any member of this family from any other, which means it cannot, on its own, tell a reader where the lamp actually is or how the floor actually sits, however exactly the shadow itself is measured.

This is not the same claim as “the recovery is imprecise” or “the recovery is ill-conditioned,” and the distinction matters enough to state before any figure is shown. An ill-conditioned recovery, of the kind a shadow edge read as a profile measures elsewhere in this collection, returns an answer whose uncertainty grows large — the true value is somewhere in a widening interval, and more or better data narrows it. A gauge freedom of the kind measured here is a different animal: every member of the family fits the photograph with exactly zero error, all of them simultaneously, so there is no sense in which more precise pixel measurements or a better camera would narrow anything at all. The ambiguity is not a matter of measurement quality; it is a mathematical fact about what one photograph of a shadow can encode, true however perfectly the photograph is taken.

What moving in the family actually does #

The clearest way to see the ambiguity is to look at a second member of the family directly, drawn completely independently, and compare it with the true configuration shown above.

Two-times-ten-to-the-minus-thirteen pixels is not “very close”; on any real camera, any real print, any real screen, it is exactly the same picture, indistinguishable by any measurement anyone could actually perform on the image. And the configuration producing it is not a small perturbation of the truth in the sense a numerical method’s rounding error would be — the lamp has moved most of a metre, the floor has tilted by several degrees, both substantial, physically obvious differences if either were viewed directly rather than through the one photograph both configurations happen to agree on perfectly.

It is worth being precise about which parts of the scene are and are not free to move in this family. The card itself is held completely fixed — its shape, its position, its own orientation are the same in every member — and only the lamp and the floor move, in a specific, coupled way that keeps the card’s own cast shadow landing on exactly the same floor points from the eye’s own point of view. This is not a family of “different scenes that happen to look similar”; it is a family built by explicit construction to leave one particular photograph invariant, which is a much stronger and much more deliberate statement than mere visual resemblance would be.

Behind the unchanging photograph: the lamp swings and the floor tilts #

Seeing several members of the family superimposed, rather than one at a time, is what makes clear that μ = 0.88 was not a special or extreme case.

The family is parameterised so that μ = 1 is the truth purely as a labelling convenience; nothing about the geometry itself privileges that value, and the picture gives no indication that the middle of this particular range is more likely than either end. A homology centred on the camera’s own eye is, formally, a projective transformation of space that fixes the eye and fixes a plane not through the eye — here, the plane at the true floor’s own height — while sliding every other point along the line joining it to the eye. Because every ray from the eye is mapped to itself by such a transformation, and a photograph records only which ray each scene point lies on rather than how far along that ray it sits, the eye cannot distinguish a point from its image under the homology. The lamp is one point subject to exactly this sliding, and the floor is one plane subject to exactly this tilting, and the picture is built entirely from rays through the eye — which is the whole reason it cannot tell the two apart.

The lamp and the floor are not moving independently of each other in this construction, and that coupling is the entire mechanism. Sliding the lamp along its own ray from the eye changes where its light strikes the card and, through the card, where the shadow falls; tilting the floor by the matched amount changes where that same shadow lands on the receiving surface, in exactly the opposite sense. A single one of those two moves — the lamp sliding with the floor held flat and fixed, or the floor tilting with the lamp held fixed — would change the picture, visibly and immediately; it is the specific pairing of the two, one move undoing what the other does to the image, that produces the invariance. This is why the family has exactly one free parameter rather than two independent ones: μ is not “how far the lamp has moved” and separately “how far the floor has tilted,” it is the single dial that moves both together in the one combination that leaves the eye’s own rays untouched.

One outside fact collapses the whole family #

The ambiguity is total from inside the picture and dissolves instantly the moment one fact from outside it is supplied.

That both quantities fall monotonically rather than merely varying is what makes either one sufficient on its own. A function that returned the same length at two different values of μ would leave a residual twofold ambiguity even after the true length was supplied; a strictly monotonic function has an inverse, so a single known value — a tape measurement of a floor mark, or an independently known post height — reads off exactly one μ and, with it, exactly one lamp position and exactly one floor tilt. Neither fact needs to come from the photograph at all: a builder’s tape measure or a post whose height was written down when it was installed is exactly the kind of information a single photograph structurally cannot supply about itself, and either one is already enough.

One picture, one length is the general statement this essay’s own collapse is one instance of: a photograph alone gives every ratio in a scene and no absolute size, and the one length that closes the gap can come from almost anywhere outside the picture — a tape measure, a known object of familiar size, a manufacturer’s specification. What is worth adding here is that the same missing-length problem, applied to this particular gauge family, closes two separate unknowns — the lamp’s own position and the floor’s own tilt — with the single supplied number, because the two were never independent in the first place; they are tied together by the one parameter μ, so pinning down μ pins down both simultaneously. A light far enough away is the same theme read from the opposite end: there a lamp’s own finite distance is what a photograph struggles to certify at all once the lamp is far enough away, where here the difficulty is not distance but a whole family of equally distant-looking possibilities standing behind one unremarkable near lamp.

A wall known to be square finds the truth by touching ninety #

A known length is one way to break the ambiguity. A second receiver, related to the first by a known geometric constraint rather than by a known distance, is another, and the way it works is worth setting out in full because it does not depend on measuring any length at all.

The reason the peak sits exactly at ninety, and nowhere else, is a fact about homologies rather than a coincidence of this particular room. A homology is a projective transformation and not, in general, an isometry — it does not preserve lengths or angles except at points or along directions its own structure happens to fix. Two planes genuinely at right angles in the true scene are carried by a homology to two planes at some other angle in general, and they come back to exactly ninety degrees only when the homology applied is the identity, which happens only at μ = 1. So a wall independently known to stand square to the floor gives a curve that touches ninety at one point and one point only, and that point is the truth — not approximately, not “closest to ninety,” but exactly, to as many decimal places as the arithmetic carries.

This method has an advantage over the length-based collapse above that is worth stating plainly: it needs no measured distance at all, only a qualitative fact about the room’s own construction. A carpenter’s or architect’s assurance that a wall was built square to the floor — a fact usually true by design in an ordinary room, and checkable without a tape measure at all — is exactly the kind of knowledge this test consumes. Where the collapse figure needed someone to have measured 1.942 m or 0.850 m at some point, the wall test needs only the kind of assumption a floor plan or a building code already states as a matter of course, which makes it the more practically available of the two routes to the truth in an ordinary interior.

A weaker fact: merely parallel is not merely square #

A weaker piece of outside knowledge — a second receiver merely known to be parallel to the first, rather than known to sit at some specific different angle — collapses the family too, but by a much smaller margin, and the difference is worth measuring rather than assuming.

The parallel receiver is not a hard refusal, and it is important not to overstate the weakness into an absolute one. A homology whose axis is not the plane at infinity does not, in general, fix parallelism between two planes passing through neither its own axis nor its own centre — so two truly parallel planes drift apart, very slightly, under every member of the family except the truth, and that drift is a real, measurable, monotonic signal rather than an exact zero. What the factor of thirteen says is that this particular fact is a much less sensitive instrument than a known right angle: a reader relying on “this wall looks roughly parallel to that one” to pin the family down is working with a signal easily lost in whatever measurement noise a real photograph carries, where a reader relying on a wall independently known to be square has a signal an order of magnitude larger to work with.

The general shape of this comparison — one known relationship collapsing an ambiguity sharply and a weaker relationship collapsing it only faintly — is the same lesson two lamps and one map drew about combining evidence from more than one source. There the strength of a constraint depended on the angle between two rays being intersected; here it depends on how far the known relationship between two receivers — an exact right angle against a mere parallelism — departs from the one case, an axis through the plane at infinity, that a homology would fix automatically. In both essays, the practical question is never merely “is there a second piece of information” but “how much leverage does this particular second piece of information actually have,” and the two are frequently very different answers.

What happens when only the lamp is wrong #

The family measured above holds the card fixed and varies the lamp and the floor together, in lockstep, so that the picture never moves. It is worth setting that carefully-balanced ambiguity beside the much more ordinary failure of un-casting a shadow through a lamp that is simply wrong, with no matching floor error to compensate for it.

This is the ordinary, expected shape of a systematic error — get one input wrong, by a known or unknown amount, and the output is wrong by a proportional, boundable amount, recoverable in principle the moment the lamp’s own position is corrected. It is also, precisely because it behaves so reasonably, the wrong model for what this essay has been measuring. A wrong lamp with the floor held at its true, correct height produces a shadow that does not match the photograph — the whole point of the 5.81 mm figure is that it is a real, detectable discrepancy an un-casting procedure could in principle be checked against. The gauge family earlier in this essay is not a lamp error of this kind at all: it is a lamp error paired with a compensating floor error, chosen so precisely that the two together leave no discrepancy whatsoever. The ordinary failure mode is loud and self-announcing; the gauge freedom is silent by construction.

The linearity in the borrowed figure’s own error is worth one further remark, because it is exactly the property the gauge family does not have. Displace the lamp alone by fifteen millimetres and the recovered outline moves by an amount proportional to that displacement — halve the error in the lamp’s assumed position and the recovered outline’s own error halves with it, which is what makes an un-casting procedure with a slightly-wrong lamp improvable: better calibration of the lamp buys a proportionally better outline. Nothing about μ in the gauge family is improvable in that sense. There is no calibration of the lamp alone, however precise, that narrows the family even slightly, because the family was built so that every member is exactly and permanently consistent with the photograph regardless of how well any single component of it is separately known.

The honest limit #

Everything measured here concerns one specific card, one specific true lamp position, and a family constructed by exactly one class of transformation — a homology centred on the camera’s own eye. It is worth being precise that this is not a claim that any pair of wrong guesses at the lamp and floor happens to compensate for each other; almost all such pairs produce a shadow visibly different from the true photograph, exactly as the borrowed figure’s 5.81 mm shows for an uncompensated lamp error. The family measured here is the specific, one-parameter set that a homology construction generates, and it is a genuine ambiguity precisely because it is not an arbitrary or generic pair of errors but a structured one, built from the one class of transformation a single eye’s rays cannot distinguish.

Nor does anything here address a scene with more than one card, or a card whose own shape is not known in advance. Two cards at different heights, both visible in the same photograph, would each constrain their own version of this family, and the two constraints together might well leave only the true configuration standing — a natural extension this essay’s own single-card measurement does not attempt. A shadow across a second object is the nearest existing measurement of a shadow crossing more than one surface at once, though it is built to answer a different question — what a receiver’s own shape does to a single shadow’s reading — rather than this essay’s question of how many independent cards it takes to pin an ambiguous lamp down. What is settled here is the minimal case: one card, one shadow, one photograph, and the exact shape and size of the ambiguity that remains once that is all there is.

What this is an instance of #

A shadow edge read as a profile measured a different failure of the identical underlying machinery: there, a swept light plane and a camera degrade toward an unusable measurement as the angle between them closes, down to an exact angle at which the recovery refuses outright. This essay’s failure is not a degradation at all — every member of the family here is recovered with perfect, unmoving confidence, because there is genuinely nothing in the picture that could ever distinguish one member from another. A vanishing conditioning number and an exact gauge freedom are two different shapes a recovery’s honest limit can take, and mistaking one for the other matters: a reader told “the residual is small” after this essay’s own family has been silently substituted for the truth would have no way to know anything was wrong at all, where a refusal at least announces itself.

The wider pattern this belongs to is named directly in this site’s own account of being determined up to something: a single photograph fixes a great deal and never everything, and the qualifier attached to what remains undetermined is not a rounding error to be waved away but the exact, structured, sometimes continuous shape of what one picture cannot see. How many shadows determine the object found the qualifier as a missing concavity no number of outlines reaches; where a shadow splits in two found it as a discrete topological fact hinging on a single tangent height; and this essay finds it as a whole family of otherwise-ordinary scenes standing behind one unremarkable photograph of a card and its shadow, told apart only by a tape measure, a known post, or a wall that happens to be square to the floor.